The information about quantum fluctuations and entanglement
concerning any particular physical quantity is fully encoded in the
mathematical description of the state. The information is quantum
in nature and cannot be fully represented or modeled by a single
real number value as in classical physics. The noncommutative
value, as an element of a noncommutative algebra, however,
encodes that full information. The explicit results on the
transformations illustrated that clearly. At least from the theoretical
point of view, the noncommutative value of an observable for
a fixed state described under a choice of frame of reference and
system of coordinates of the phase space, is completely and
definitely determined. When a number of observables satisfying
a certain algebraic relations as dynamical variables, that exact
relation is preserved among their noncommutative values for
any particular fixed state, like the (real number) values of the
classical observables. Under a quantum reference frame
transformation, the state description is changed. The mathematical
description of any particular physical quantity of the state has hence
to be changed. Unlike a classical reference frame transformation
which only changes the expectation value of the observable, the
quantum transformation may change every aspect of a state. Such
a change as given by the corresponding noncommutative value
again encodes the full information involved, from which one can
also read off the changes in the quantum fluctuations and
entanglement. Without the use of the notion of the noncommutative
values, changes in the description of any particular physical properties
under a quantum reference frame transformation are only given as
state independent observable relations [1]}, [2]}. Tracing how some
states change under the transformation, as for example presented
in figure  3 of Ref.[1]} can illustrate the very interesting features.
The noncommutative values calculations answer all those explicitly.
The picture also helps to demonstrate that our newly introduced
notion of the noncommutative values at least serves as a very useful
model for the `value' of the quantum observables as an exact, hence
mathematically abstract, description of the corresponding physical
properties for specific quantum states.
